Three resistors each of 2 ohm are connected together in a triangular shape. The resistance between any two vertices will be

Options

A. `4//3` ohm
B. `3//4` ohm
C. 3 ohm
D. 6 ohm

Answer

To find the resistance between any two vertices of a triangular arrangement of three resistors, each of 2 ohms, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have three resistors (R1, R2, R3) each with a resistance of 2 ohms connected in a triangular shape. Let's label the vertices of the triangle as A, B, and C. The resistors are connected as follows: R1 between A and B, R2 between B and C, and R3 between C and A.

2. **Identify the Points of Interest**:
- We need to find the equivalent resistance between any two vertices. For this solution, let's calculate the resistance between points A and B.

3. **Analyze the Circuit**:
- When measuring the resistance between points A and B, the resistors R2 (between B and C) and R3 (between C and A) are in parallel. The resistor R1 (between A and B) is in series with the equivalent resistance of R2 and R3.

4. **Calculate the Equivalent Resistance of R2 and R3**:
- The formula for two resistors in parallel (R2 and R3) is given by:
\[ R_{parallel} = \frac{R2 \cdot R3}{R2 + R3} \]
- Substituting the values:
\[ R_{parallel} = \frac{2 \, \Omega \cdot 2 \, \Omega}{2 \, \Omega + 2 \, \Omega} = \frac{4 \, \Omega^2}{4 \, \Omega} = 1 \, \Omega \]

5. **Combine with R1**:
- Now, we add the equivalent resistance of the parallel combination (1 ohm) to R1 (2 ohms):
\[ R_{AB} = R1 + R_{parallel} = 2 \, \Omega + 1 \, \Omega = 3 \, \Omega \]

6. **Final Result**:
- The resistance between any two vertices (A and B) is 3 ohms.